Elena A. Lebedeva

Periodic wavelet frames and time–frequency localization

Abstract A family of Parseval periodic wavelet frames is constructed. The family has optimal time–frequency localization (in the sense of the Breitenberger uncertainty constant) with respect to a family parameter and it has the best currently known localization with respect to a multiresolution analysis parameter.

Wavelet frames on Vilenkin groups and their approximation properties

An explicit description of all Walsh polynomials generating tight wavelet frames is given. An algorithm for finding the corresponding wavelet functions is suggested, and a general form for all wavelet frames generated by an appropriate Walsh polynomial is described. Approximation properties of tight wavelet frames are also studied. In contrast to the real setting, it appeared that a wavelet tight frame decomposition has an arbitrary large approximation order whenever all wavelet functions are compactly supported.

Uncertainty constants and quasispline wavelets

Abstract In 1996 Chui and Wang proved that the uncertainty constants of scaling and wavelet functions tend to infinity as smoothness of the wavelets grows for a broad class of wavelets such as Daubechies wavelets and spline wavelets. We construct a class of new families of wavelets (quasispline wavelets) whose uncertainty constants tend to those of the Meyer wavelet function used in construction.

Meyer wavelets with least uncertainty constant

In the present paper, we construct a system of Meyer wavelets with least possible uncertainty constant. The uncertainty constant minimization problem is reduced to a convex variational problem whose solution satisfies a second-order nonlinear differential equation. Solving this equation numerically, we obtain the desired system of wavelets.

Computational implementation of the inverse continuous wavelet transform without a requirement of the admissibility condition

Recently, it has been proven [R. Soc. Open Sci. 1 (2014) 140124] that the continuous wavelet transform with non-admissible kernels (approximate wavelets) allows for an existence of the exact inverse transform. Here we consider the computational possibility for the realization of this approach. We provide modified simpler explanation of the reconstruction formula, restricted on the practical case of real valued finite (or periodic/periodized) samples and the standard (restricted) Morlet wavelet as a practically important example of an approximate wavelet. The provided examples of applications includes the test function and the non-stationary electro-physical signals arising in the problem of neuroscience.

On alternative wavelet reconstruction formula: a case study of approximate wavelets.

The application of the continuous wavelet transform to the study of a wide class of physical processes with oscillatory dynamics is restricted by large central frequencies owing to the admissibility condition. We propose an alternative reconstruction formula for the continuous wavelet transform, which is applicable even if the admissibility condition is violated. The case of the transform with the standard reduced Morlet wavelet, which is an important example of such analysing functions, is discussed.

Minimization of the Uncertainty Constant of the Family of Meyer Wavelets

We obtain a simplified expression for the uncertainty constant of a Meyer wavelet. Using this expression, we find the lower bound of the uncertainty constant and construct the Ritz minimizing sequence.

A generalization of Jentzsch’s theorem

We study the asymptotic behavior of the roots of polynomials given by a linear summation method for partial sums of the Fourier series.

Localization of functions defined on cantor group

We introduce a notion of localization for dyadic functions, i. e. functions defined on Cantor group. Both non-periodic and periodic cases are discussed. Localization is characterized by functionals UCd and UCdp similar to the Heisenberg (the Breitenberger) uncertainty constants used for real-line (periodic) functions. We are looking for dyadic analogs of uncertainty principles. To justify definition we use some test functions including dyadic scaling and wavelet functions.

A directional uncertainty principle for periodic functions

In this paper we introduce a notion of a directional uncertainty product for multivariate periodic functions and multivariate discrete signals. It measures a localization of a signal along a particular direction. We study properties of the uncertainty product and give an example of well localized multivariate periodic Parseval wavelet frames.